To find the value of [tex]\(\tan(x + y)\)[/tex] given the equation [tex]\(3 \sin x + 12 \sin y + 4 \cos x + 5 \cos y = 18\)[/tex], follow these steps:
1. Understand the problem: We are given a trigonometric equation and need to find the value of [tex]\(\tan(x + y)\)[/tex].
2. Equation Analysis: Analyze the given trigonometric equation:
[tex]\[
3 \sin x + 12 \sin y + 4 \cos x + 5 \cos y = 18
\][/tex]
3. Trigonometric Identity: Recall the trigonometric identities:
- [tex]\(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)[/tex]
- [tex]\(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)[/tex]
- [tex]\(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\)[/tex]
4. Finding [tex]\(\tan(x + y)\)[/tex]: We need to determine [tex]\(\tan(x + y)\)[/tex].
5. Solution Conclusion: In this problem, without further specific numerical values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], you can determine from the provided problem details that the value of [tex]\(\tan(x + y)\)[/tex] is:
[tex]\[
\tan(x + y) = -63
\][/tex]
Thus, the answer is [tex]\(\tan(x + y) = -63\)[/tex].
